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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00017901534272889592

   1. Initial program 0.1

      \[\frac{e^{x} - 1}{x}\]

   2. Initial simplification0.1

      \[\leadsto \frac{-1 + e^{x}}{x}\]
   3. Using strategy rm

   4. Applied flip-+0.1

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot -1 - e^{x} \cdot e^{x}}{-1 - e^{x}}}}{x}\]

   5. Applied associate-/l/0.1

      \[\leadsto \color{blue}{\frac{-1 \cdot -1 - e^{x} \cdot e^{x}}{x \cdot \left(-1 - e^{x}\right)}}\]

   6. Simplified0.1

      \[\leadsto \frac{\color{blue}{1 - e^{x} \cdot e^{x}}}{x \cdot \left(-1 - e^{x}\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt0.1

      \[\leadsto \frac{1 - e^{x} \cdot e^{x}}{x \cdot \color{blue}{\left(\left(\sqrt[3]{-1 - e^{x}} \cdot \sqrt[3]{-1 - e^{x}}\right) \cdot \sqrt[3]{-1 - e^{x}}\right)}}\]

   if -0.00017901534272889592 < x

   1. Initial program 59.9

      \[\frac{e^{x} - 1}{x}\]

   2. Initial simplification59.9

      \[\leadsto \frac{-1 + e^{x}}{x}\]

   3. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]
   4. Using strategy rm

   5. Applied associate-+r+0.5

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + \frac{1}{6} \cdot {x}^{2}\right) + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00017901534272889592:\\ \;\;\;\;\frac{1 - e^{x} \cdot e^{x}}{\left(\sqrt[3]{-1 - e^{x}} \cdot \left(\sqrt[3]{-1 - e^{x}} \cdot \sqrt[3]{-1 - e^{x}}\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot {x}^{2} + x \cdot \frac{1}{2}\right) + 1\\ \end{array}\]

Runtime

Time bar (total: 11.1s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))